American Institute of Mathematical Sciences

Advanced
September  2019, 14(3): 589-615. doi: 10.3934/nhm.2019023

Optimal reinforcing networks for elastic membranes

Giovanni Alberti 1, , Giuseppe Buttazzo 1,, , Serena Guarino Lo Bianco 2, and  Édouard Oudet 3,

1. 

Dipartimento di Matematica, Università di Pisa, l.go B. Pontecorvo 5, 56127 Pisa, Italy
2. 

Dipartimento di Matematica e an, 80126 Napoli, Italy
3. 

Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 38041 Grenoble, France

* Corresponding author

Received  August 2018 Revised  March 2019 Published  May 2019

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal network, and observe that such network carries a multiplicity that in principle can be strictly larger than one. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when the total length becomes large.

Keywords: Optimal networks, elastic membranes, reinforcement, relaxed solution, Golab's semicontinuity theorem.
Mathematics Subject Classification: 49J45, 49Q10, 35R35, 35J25, 49M05.
Citation: Giovanni Alberti, Giuseppe Buttazzo, Serena Guarino Lo Bianco, Édouard Oudet. Optimal reinforcing networks for elastic membranes. Networks & Heterogeneous Media, 2019, 14 (3) : 589-615. doi: 10.3934/nhm.2019023
References:
[1]

G. Alberti and M. Ottolini, On the structure of continua with finite length and Golab's semicontinuity theorem, Nonlinear Anal., 153 (2017), 35-55.  doi: 10.1016/j.na.2016.10.012.  Google Scholar

[2]

E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math., 386 (1988), 99-115.  doi: 10.1515/crll.1988.386.99.  Google Scholar

[3]

M. Beckmann, A continuous model of transportation, Econometrica, 20 (1952), 643-660.  doi: 10.2307/1907646.  Google Scholar

[4]

G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations, 5 (1996), 37-54.  doi: 10.1007/s005260050058.  Google Scholar

[5]

L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671.  doi: 10.1016/j.matpur.2010.03.010.  Google Scholar

[6]

G. Buttazzo, G. Carlier and S. Guarino Lo Bianco, Optimal regions for congested transport, ESAIM Math. Model. Numer. Anal., 49 (2015), 1607-1619.  doi: 10.1051/m2an/2015022.  Google Scholar

[7]

G. Buttazzo, É. Oudet and B. Velichkov, A free boundary problem arising in PDE optimization, Calc. Var. Partial Differential Equations, 54 (2015), 3829-3856.  doi: 10.1007/s00526-015-0923-1.  Google Scholar

[8]

G. Buttazzo, É. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002, 41–65.  Google Scholar

[9]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777.  doi: 10.3934/nhm.2007.2.761.  Google Scholar

[10]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769.  doi: 10.1051/cocv:2006020.  Google Scholar

[11]

G. Buttazzo and N. Varchon, On the optimal reinforcement of an elastic membrane, Riv. Mat. Univ. Parma (Ser. 7), 4 (2005), 115-125.   Google Scholar

[12]

Y.-H. Dai and R. Fletcher, New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. (Ser. A), 106 (2006), 403-421.  doi: 10.1007/s10107-005-0595-2.  Google Scholar

[13] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.   Google Scholar
[14]

S. Golab, Sur quelques points de la théorie de la longueur, Ann. Soc. Polon. Math., 7 (1929), 227-241.   Google Scholar

[15]

S. G. Johnson, The NLopt nonlinear-optimization package., Available from: http://ab-initio.mit.edu/nlopt. Google Scholar

[16]

S. J. N. Mosconi and P. Tilli, Γ-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.   Google Scholar

[17]

E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[18]

J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, 1 (1952), 325-362.  doi: 10.1680/ipeds.1952.11362.  Google Scholar

show all references

References:
[1]

G. Alberti and M. Ottolini, On the structure of continua with finite length and Golab's semicontinuity theorem, Nonlinear Anal., 153 (2017), 35-55.  doi: 10.1016/j.na.2016.10.012.  Google Scholar

[2]

E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math., 386 (1988), 99-115.  doi: 10.1515/crll.1988.386.99.  Google Scholar

[3]

M. Beckmann, A continuous model of transportation, Econometrica, 20 (1952), 643-660.  doi: 10.2307/1907646.  Google Scholar

[4]

G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations, 5 (1996), 37-54.  doi: 10.1007/s005260050058.  Google Scholar

[5]

L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671.  doi: 10.1016/j.matpur.2010.03.010.  Google Scholar

[6]

G. Buttazzo, G. Carlier and S. Guarino Lo Bianco, Optimal regions for congested transport, ESAIM Math. Model. Numer. Anal., 49 (2015), 1607-1619.  doi: 10.1051/m2an/2015022.  Google Scholar

[7]

G. Buttazzo, É. Oudet and B. Velichkov, A free boundary problem arising in PDE optimization, Calc. Var. Partial Differential Equations, 54 (2015), 3829-3856.  doi: 10.1007/s00526-015-0923-1.  Google Scholar

[8]

G. Buttazzo, É. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002, 41–65.  Google Scholar

[9]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777.  doi: 10.3934/nhm.2007.2.761.  Google Scholar

[10]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769.  doi: 10.1051/cocv:2006020.  Google Scholar

[11]

G. Buttazzo and N. Varchon, On the optimal reinforcement of an elastic membrane, Riv. Mat. Univ. Parma (Ser. 7), 4 (2005), 115-125.   Google Scholar

[12]

Y.-H. Dai and R. Fletcher, New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. (Ser. A), 106 (2006), 403-421.  doi: 10.1007/s10107-005-0595-2.  Google Scholar

[13] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.   Google Scholar
[14]

S. Golab, Sur quelques points de la théorie de la longueur, Ann. Soc. Polon. Math., 7 (1929), 227-241.   Google Scholar

[15]

S. G. Johnson, The NLopt nonlinear-optimization package., Available from: http://ab-initio.mit.edu/nlopt. Google Scholar

[16]

S. J. N. Mosconi and P. Tilli, Γ-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.   Google Scholar

[17]

E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[18]

J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, 1 (1952), 325-362.  doi: 10.1680/ipeds.1952.11362.  Google Scholar

Figure 1.  Approximation of globally optimal reinforcement structures for $ m = 0.5 $, $ L = 1, \, 2 $ and $ 3 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Approximation of globaly optimal reinforcement structures for $ m = 0.5 $, $ L = 4, \, 5 $ and $ 6 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Approximation of globaly optimal reinforcement strucutres for m = 0.5, L = 1.5, 2.5 and 5 for a source consisting of two dirac masses. The upper colorbar is related to the weights θ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Reinforcement values computed on a fine mesh of $ 10^6 $ elements for classical and computed connected sets for $ m = 0.5 $
Length constraint Theoretical guesses Computed optimal networks
1 -0.179471 (radius) -0.178873
2 -0.165095 (diameter) -0.161944
3 -0.152676 (star) -0.149601
4 -0.141969 (cross) -0.138076
5 - -0.127661
6 - -0.117140
Table Options

Download as excel

Length constraint Theoretical guesses Computed optimal networks
1 -0.179471 (radius) -0.178873
2 -0.165095 (diameter) -0.161944
3 -0.152676 (star) -0.149601
4 -0.141969 (cross) -0.138076
5 - -0.127661
6 - -0.117140
[1]

J. G. Simmonds. The hypercircle theorem for elastic shells and the accuracy of Novozhilov's simplified equations for general cylindrical shells. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 643-650. doi: 10.3934/dcdsb.2007.7.643
[2]

Armando G. M. Neves. Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 611-624. doi: 10.3934/cpaa.2010.9.611
[3]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619
[4]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011
[5]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367
[6]

Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225
[7]

Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303
[8]

Kenji Kimura, Jen-Chih Yao. Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 167-181. doi: 10.3934/jimo.2008.4.167
[9]

Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
[10]

Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313
[11]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
[12]

Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
[13]

Johan Henriksson, Torbjörn Lundh, Bernt Wennberg. A model of sympatric speciation through reinforcement. Kinetic & Related Models, 2010, 3 (1) : 143-163. doi: 10.3934/krm.2010.3.143
[14]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[15]

Zhong-Jie Han, Enrique Zuazua. Decay rates for elastic-thermoelastic star-shaped networks. Networks & Heterogeneous Media, 2017, 12 (3) : 461-488. doi: 10.3934/nhm.2017020
[16]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413
[17]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
[18]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[19]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945
[20]

Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (53)
  • HTML views (231)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences